ada 114535
TRANSCRIPT
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AD-A1IN
535
WISCONSIN UNIV-MADISON
MATHEMATICS RESEARCH
CENTER
F/G 12/1
MATHEMATICAL
MODELINO OF TWO-PHASE
FLOW.(U)
MAR 82
D A
DREW
DAAG29-8o-C-O00I
UNCLASSIFIED
MRC-TSR-2343
NL
Ei
IIIIIIIIIIII
IIIIIIIIIIIIIu
EIIIIIIEIIIIIE
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MRC Technical
Summary
Report
#2343
MATHEMATICAL
MODELING
OF TWO-PHASE
FLOW
S
D. A.
Drew
Mathematics
Research
Center
University
of Wisconsin-Madison
610
Walnut
Street
Madison,
Wisconsin
53706
March
1982
(Received
February
11,
1982)
Approved
for
public
release
D
T
I
Cby
Distribution
unlimited
,E
T
L.LU.S.
Army Research Office
P.
0.
Box
12211
Fsearch Triangle
Park
rth Carolina
27709
8
028
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UNIVERSITY OF WISCONSIN
-
MADISON
MATHEMATICS
RESEARCH CENTER
MATHEMATICAL MODELING OF TWO-PHASE
FLOW
D. A. Drew
Technical
Summary
Report #
2343
March
1982
ABSTRACT
A continuum mechanics
approach to
two-phase
flow
is reviewed.
An
averaging
procedure is discussed
and
applied
to
the exact equations
of
motion.
Constitutive
equations are
supplied and discussed for
the
stresses,
pressure
differences
and the
interfacial force. The
nature
of
the resulting
equations is
studied.
AMS
(MOS) Subject
Classification: 76T05
Key Words:
Two-Phase Flow,
Constitutive equations
Work
Unit
Number
2
-
Physical
Mathematics
Spon-;ored by
the United States
Army under
Contract No.
DAAG29-80-C-041.
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SIGNIFICANCE
AND
EXPLANATION
Equations describing
the
motions
of
two materials,
one
dispersed
throughout
the other,
are derived.
There
is some
controversy
over
the
ability
of certain
simplified
forms
of these
equations
to predict
meaning-
ful flows.
Consequently,
evidence
is
presented
to
verify
the forms
of
various
terms
appearing
in the equations.
The consequences
of
the assumed
forms
are discussed.
Accesston
For
NTIS
GRA&I
DTIC
TAB
Unannounced
Justification
By- -
Distribution/
Availability
Codes
'Avail
and/or
TI
Dist
Special
CP
The
responsibility
for
the
wording
and views
expressed
in this
descriptive
summary
lies with MRC,
and not
with
the author of
this report.
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I.
MATHEMATICAL
MODELING
OF
TWO-PHASE
FLOW
D. A.
Drew
INTRODUCTION
Dispersed
two-phase
flows
occur
in
many
natural and technological
situations.
For
example,
dust
in
air and
sediment
in
water contribute
to
ernsion
and
silting, and can cause problems for machinery
such as
helicopters and
power plants operating
in such
an environment.
Also,
many energy
conversion and
chemical
processes
involve two-phase flows.
Boiling
a fluid such as
water or
sodium and extracting
the heat
by
condensation
at a
different
location
provides a practical
energy flow
process. Many
kinds
of
chemical
reactants and
catalysts
are
mixed
in
a
dispersion of fine
particles to expose
as much
interfacial area
as
possible.
In
such a large class
of
problems,
many
diverse mechanisms
are
important
in
different situations.
The
models used for
these
different
situations have
many common
features,
such as
interfacial drag.
This
paper
examines
the
common
features of
dispersed
two phase
flows
from a continuum mechanical approach. Since
it is
not
universally
accepted
that such an
approach is
valid, we shall
discuss
some
philosophical
reasons
for taking
such
an
approach. The approach
is
based on the
view that it is
sufficient
to describe
each
material
as
a
continuum,
occupying
the same
region
in
space.
This new
"material"
consists of
two
interacting
materials (called
phases, even
though they
often
are not
different
phases of the
same material).
The two phase
material
is
often called
the
mixture.
In
analogy with continuum
mechanics,
we shall have
to
specify
how the
mixture interacts
with
itself.
In
ordinary continuum
mechanics
(ignoring thermodynamic
Sponsored by
the
United States
Army under Contract
No.
DAAG29-80-C-0041.
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considerations) that
requires
a
constitutive
equation expressing
the
stress
as a function of
various fields.
In two-phase
flow
mectianics, it
requires
specification
of stresses for
each
phase, plus a
relation for
the
interaction
of the two materials.
Researchers who
do not subscribe
to this
view treat
the mechanics
of
the
two
materials, plus
the dynamics
of the interface
as fundamental,
and use
derived
results
and/or
measurements
to
gain
the understanding
needed for
their particular
application. It
is instructive
to consider
the
approaches to gas
dynamics.
Most scientists believe
that
a gas
is a
collection
of many molecules
which move,
vibrate, and
interact
in
a
complex,
but describable
way. Indeed,
with the
aid of
large computers,
molecular
dynamics has made
great strides
in
understanding such
phenomena
as shocks and
phase
transition,
among
others. In spite
of the
knowledge
of the correctness
of this
model, many scientists
and
engineers use a
continuum model
for gas
dynamics.
Indeed,
anyone
wishing
to describe, for
example,
the flow
around
an airfoil, would
be
hard
pressed
to find a
computer big enough
to do it as
a problem
in
molecular dynamics. As
a
problem in continuum
gas
dynamics, it is
still
a
large problem;
however,
it is done
quite
routinely
numerically,
including shocks.
In addition,
certain solutions
to
the
continuum
equations can
be
obtained analytically.
While these are
not always of
direct technical
interest,
they often
suggest
phenomena
or techniques
which
do have
direct
bearing on problems
of
interest. Experiments
in
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gas
dynamics can be useful. Without a model for
comparison
or
scaling,
however, one is limited
to
understanding
only
the
particular
geometry
and scale of
the
experiment.
The
ability to infer
is
lost.
While
these
points seem obvious
for
gas dynamics,
they
nevertheless
should be
discussed
and understood for the
analogous situation
of two-
phase flow mechanics.
For a
particular
flow with ,nly
a
few
particles
which
interact
only slightly,
it is
best (and
perhaps even
necessary) to
describe
them
molecularly ,
that is, by
predicting
the
trajectories of
each one. If many particles are involved, it is
better
to
use a
continuum
description.
As in
gas
dynamics, both descriptions have their
place.
Furthermore,
it
is
often
instructive to try
to
ascertain
what
one model implies
about
the
other,
as in using the Boltzmann
description
of
a
gas
to get
continuum properties.
The
analogy
is apt for
a continuum description
of
two-phase
flows. This paper will review the
connection
with the
exact, or
microscopic description
through the
application
of
an
averaging process
to the
continuum mechanical
equations
describing
the exact motion
of
each material
at
each
point. If the
exact
flows
were
known,
the
averaged
equations would be
completely determined,
and
therefore
unnecessary, since
desired
averaged information
(such
as the
average
concentration of particles) could
be
determined without using the
averaged
equations.
The solutions are not
known,
nor is it necessary to
the continuum approach that they
be
known.
The
resulting averaged
eluations
are assumed to describe a
material
(the mixture) for which the
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interactions
must be specified. This specification
is
then
done
according
to
certain
rules
which
are
reasonable and not
too
limiting.
We
shall
discuss these
rules and their
implications.
The forms
of the resulting
equations
are determined by the
choice
of
a list
of
variables
which
are
assumed
to
influence the
interactions.
The
resulting
equations have several
unknown
coefficients.
These
coefficients
are assumed to be
determinable
by experiments.
For the
model
we propose, we
examine the
experimental data
and their
implications
on
the
coefficients.
The result
is, in
essence, a
recommendation for
a model which
has
many
known
features of
two-phase
flow dynamics.
An
enlightened
investigator can
use the
model
to
make
predictions
in
a
situation which
falls
within
the
range of
assumptions
made.
The model
can also be
a starting point
to obtain generalizations
(such
as
the
inclusion of
electromagnetic effects).
As with
all models,
it
should never
be
used blindly, but
with caution and
careful
examination
of the
results and
implications.
Historically,
geophysical
flows
involving
sediment
and
clouds were
among the
first two-phase
flows observed from
a scientific
point
of
view. Sediment
meant
erosion and loss of
property, or deposition
and
loss
of navigability.
Clouds signalled
possible
rainfall. The
amounts
could
mean
drought,
bountiful
harvests
or
devastating
floods.
An
analytical
description
awaited the
development
of
the
calculus,
fluid
mechanics
and
partial differential equation
theory. The
industrial
revolution spurred the need
for
understanding of all of
basic science,
including two phase
flows, although the
correlation
between
progress and
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the
acquisition
of
fundamental knowledge
is not exact.
For
example,
automobiles
have
worked
reasonably
well
without
a detailed knowledge of
the
flow
and
evaporation of
fuel droplets; on the
other
hand, efficient
design
of fluidized beds
have been reliant on
the understanding and
stabilizing effects obtained
from
analytical
models.
Early work on the
form taken by beds of
particles subject to forces
due to
flowing fluid (DuBuat 1786,
Helmholtz
1888,
Blasius
1912
and
Exner 1920)
lead to interesting
fundamental
results
such as the
stability
of
the
interface between
two fluids
and a
practical
understanding
of the
macroscopic
properties of
sedimentation
and bed
form
evolution.
A more
microscopic
look
(Bagnold 1941)
dealt
with
the
mechanics
of the
interactions between
the
particles
and
the
fluid, and
the particles
and
the
bed. A
recent
perspective
is
given
by
Engelund
and Freds~e
(1982).
Developments in
cloud physics occured in physical
chemistry
of
nucleation and formation.
The
need
for
detailed
mechanical
considerations was low
due to
the
fact that
the
vapor,
nucleation sites
and the
formed droplets
all
flow with the surrounding air
(until
raindrops
form).
Porous
medium
theory, a two phase flow where
the solid
phase does
not
move
appreciably,
developed with some
different concerns (Darcy
1856), but Blot
(1955) was
instrumental
in putting
the empirical
knowledge
on
a sound continuum
basis. The
desire
to
extract
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hydrocarbons
from
deep inside the earth's crust has given
a
great
impetus to
the
study
of
flow in porous media.
See Scheidegger's (1974)
book for a discussion of porous
bed modeling concerns.
The next important milestone
in
the
development of two-phase flow
continuum
mechanics theory was the development and use of equations of
dusty gases
in
the
1950's.
Models for dusty gases are summarized
by
Marble (1970).
Chemical processing in fluidized beds gave an urgency to the
development
of the theory of particle-fluid systems.
Early theoretical
papers
include those
of Jackson
(1963), Murray (1965a,b),
and Anderson
and Jackson
(1967, 1968).
The flow regime
involved in fluidization is
one
of the most
difficult for particle-fluid flows. The
particle
concentrations are high,
the dispersed phase
is
relatively
dense,
the
dispersed
phase
undergoes
a
random
micromotion, and
often
is is
desirable to
have chemical reactions
occur in the
flow.
In
the
early 1960's
the
emergence
of commercial atomic
energy
spurred
the study
of flows of steam and
water.
The
work
of Zuber (1964)
was a
pioneering landmark.
Fluid-fluid
flows have a difficulty
not
encountered in particle-fluid flows, namely that the shape
of the
dispersed-phase can change, resulting in
changing interfacial
area
and
consequent interactions between the fluids.
In
spite
of much progress
(Lahey & Moody
1977),
two phase flow studies
in nuclear
reactors are
still
a
concern.
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The parallel development
of
mixture
theory
had some
small
influence
on
the
progress
of
two-phase flows. Indeed,
the
concept
of
interpenetrating
continuum
is
natural
in
mixtures where the
dispersion
occurs on the
molecular level.
The development
of
the
ideas of
diffusion by
Fick
(1855),
the
thermodynamic
concepts in
mixtures by
Duhem (1893),
Meixner (1943),
Prigogine
and
Mazur (1951)
lead naturally
to the theory
of mixtures
expounded by
Truesdell and Toupin
(1960).
The
theory
of
mixtures,
as applied to
the specific
mixtures
where
the two
constituents remain
unmixed,
is
the
basis of the present
work, and
(whether
explicitly
recognized or not)
much of the more specific
previous
work. Kenyon
(1976)
applies
the
mixture theory
to multiphase
flows. See
also the
review
by
Bedford and
Drumheller (1982).
By
the
1960's enough of the common
features of the
diverse two-
phase flows
were
evident. Several
influential books
of a general nature
appeared, including
Fluid
Dynamics of Multiphase
Systems by Soo (1967),
One-dimensional
Two-Phase
Flow, by Wallis (1969),
and Flowin
Gas-Solid
Susensions,
by
Boothrcyd (1971).
Wallis' approach
was
strongly
influenced by
gas-liquid
flows,
and
dealt
with
the basic concept
of
interpenetrability by
considering
cross-sectionally
averaged
equations,
and
introduced
constitutive
assumption
by
quoting appropriate
experiments. Soo's work
was largely
based on
particle-fluid
flows. He
assumed
interpenetrability from
the
start,
and
included
forces
in
the
particular momentum
equation known
from experiments or
inferred
from
calculations.
Boothroyd was
interested
in
particle-gas flows,
and
contributed ideas on
turbulence
and
drag reduction.
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A different, but
somewhat related approach
to the problem of
a
mechanical description of two-phase flows
uses the
single-particle or
several interacting particle
flow
fields, along
with
an averaging
approach
to yield rheological
or
transport properties.
The
celebrated
result
in this area is the
effective
viscosity result due to
Einstein
(1906),
which shows
that in
slow
flow the
mixture behaves like a fluid
5
with viscosity
increased by a factor of 1
+
,
where a
is
the
2
volumetric concentration
of
particles. An approach outlined
by
Brinkman
(1947)
has been
influential.
The
philosophical roots of
this
approach
are elegantly
discussed
in the
book LowReynolds Number H drod
namics,
by Happel and Brenner (1965).
The idea
is to
use solutions of
the flow
equations in
special
cases (such as
Stoke's
flow around an
array of
spheres) to infer
information about the flow
(such
as the
total force on
the
particulate
phase).
Several aspects
of this
work have been
discussed
in
this Review; see the
papers
by
Brenner
(1970),
Batchelor
(1974), Herczynski
and
Pienkowska
(1980),
Leal (1980) and Russel
(1981).
Several authors (Bedford
& Drumheller
1978,
Drumheller
& Bedford
1980) have
pursured a variational approach to two-phase
flows,
generalizing the work of
Biot
(1977).
The variational
formulation
starts with
Hamilton's principle,
given
by
t
I
t
2
ft (T-V)dt
f2
6Wdt
=0
t6d
0
where
6 represents the variation
over an
appropriate
space
of
functions,
T
and
V
are
the
kinetic
and
potential energies,
6W
is
the virtual work and
t
1
and t
2
are
two arbitrary
times. A
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variational formulation has
an
advantage
in
that if it
is desired
to
include
a
certain effect
(for
example,
the
virtual
work), that
effect
would
be included
consistently in the
mass,
momentum, and
kinetic energy
equations.
A
more concrete
and immediate
advantage is
in
formulating
numerical techniques, and
specifically, finite element techniques,
where
variational formultion
eases the
translation of the
partial
differential
equations into discrete
equations.
In
order
to use
a variational formulation,
it is
necessary to
define
the
variation 6,
the
kinetic
and
potential energies T
and
V
and the
virtual work 6w.
Constraints must
be
included with
Lagrange
multipliers. Bedford
& Drumheller (1982)
note that
Hamilton's
principle
is
usually
formulated with a
control mass , that is, for
material
volumes. The
problem
of
having
two materials
moving
at
different
velocities
is
dealt with by assuming
a control volume which
coincides
with
a
rigid
surface
through
which no material of either
phase passes.
They use the technique to
show how
the
effect
of
oscillations in
bubble
diameter
can
be included by
including
the
kinetic energy of the
liquid due to
a change in
bubble
diameter. They similarly
include
virtual
mass by arguing
that relative
accelerations
increase
the kinetic
energy.
A conceptual
difficulty with the
approach
is
that
the
user must
decide
which
fluctuations
contribute to the
total
kinetic
energy,
and
which
contribute
to
virtual work.
For
example, viscous drag is
due
to
fluctuations of
the
fluid
velocity
near a
particle,but
is
included
as
a
virtual work
term. A
rule
of
thumb might be
to consider
whether the
energy is recover-ible
or not.
Virtual work
is
associated with
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unrecoverable
energy,
kinetic
and
potential
energy
with recoverable
energy.
Variational
formulations
(in general)
do not
mention
the effect
of
Reynolds stresses, which
are one
manifestation
of
fluctuations.
We
conclude
that
while
variational
formulations
are
useful,
it is
not
always
straightforward
to
formulate
them.
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EQUATIONS OF MOTION
We start
by
assuming
that each material
involved can
be
described
as
a continuum, governed
by the
partial differential equations of
continuum mechanics. The materials
are separated by
an
interface, which
we
assume
to be a
surface. At
the
interface,
jump conditions
express
the conditions of conservation of mass and momentum.
The equations of motion
for
each phase are (Truesdell
and
Toupin
1960)
(1)
conservation
of mass
ap
+
V-PV
= 0
(1)
at
(2) conservation of linear momentum
aPV +
V-pvv =
V.T
+ pf
(2)
at
valid in the interior of each phase. Here P denotes
the
density, w
the
velocity, T
the
stress tensor, and f
the
body force density.
Conservation of angular momentum
becomes
T = Tt, where t
denotes
the
transpose. At
the interface, the jump
conditions
are
(1) jump
condition
for
mass
[[P(v - v.) 'n]] =
0 (3)
1
(2) jump condition
for
momentum
[[Pv(v -
v )n-
T'n]]
= aKn
. (4)
Here [[
]
denotes
the
jump across the interface, v
i
is that
velocity
of the
interface,
a is the surface tension, assumed to be a
constant, K is the
mean curvature
of the interface, and n is the
unit normal (Aris 1962). We shall assume
that
n points
out oF
phase
k, and that the jump between f in phase k
and
f in
phase
I is
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k
defined by
([f]] = f _
f , where
a
superscript
k denotes
the
limiting value from
the
phase
k side.
As
a
sign
convention for the
curvature,
we
assume
that
K
is
positive (concave)
toward
-n.
The
mass
of
the interface
has been
neglected,
and
the surface stresses
have
been assumed
to be
in
the form of
classic
surface
tension.
We do not
discuss any
thermodynamic
relations in this
paper.
Thermodynamic
considerations are
important for
many
multiphase flows.
Our discussion focuses
on
the mechanics of two-phase
flows, hence we
elect to
forego a discussion of
thermodynamics for
the sake
of
simplicity.
Constitutive
equations
must be
supplied
to
describe the
behavior
of
each
material
incolved. For
example, if one
material is
an
incompressible liquid,
then
specifying
the value
of
p,
and
assuming
T =
-pI + p(Vv
+
(Vv)
determines the nature of the
behavior of the
fluid in
that
phase.
Similar considerations
are
possible for solid
particles or a
gas.
The resulting
differential
equations,
along with
the
jump
conditions,
provide
a fundamental description
of the
detailed
or exact
flow.
Usually,
however, the
details of
the
flow
are
not required.
For
most
purposes of equipment or
process
design, averaged, or
macroscopic
flow information
is
sufficient.
Fluctuations,
or
details in the flow
must be
resolved only to
the extent that
they effect
the mean flow
(like
the Reynolds stresses
effect the
mean flow in
a turbulent
flow).
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Averaging
In order
to
obtain
equations which
do not
contain
the details
of
the
flow,
it
has
become customary to
apply some sort of averaging
process.
It
is not essential
to do
so; indeed,
some researchers prefer
to postulate macroscopic equations without reference
to any microscopic
equations.
Certainly, the necessary
terms in
the
macroscopic equations
can
be
deduced
without using
an
averaging process.
One advantage
of
a
postulational
approach
is
obvious
- not having
to deal with
the
worries
of
the averaging process. The
advantages of
averaging are
less obvious.
First, the various
terms appearing
in the macroscopic equations are
shown
to arise
from
appropriate
microscopic
considerations. For
example, stress terms arise from
microscopic stresses (pressure, for
example) and also from
velocity fluctuations
(Reynolds stresses).
Knowledge of
this fact does
give a modicum of insight into
the
formulation of
constitutive equations. If
a
term
appeared
in
the
averaging
which
was not
expected in the postulational
approach,
then
it
would be obligatory
to
include
it
in the
postulated
model,
or else
explain why it is
superfluous.
An
additional advantage
of
averaging is that
the
resulting
macroscopic
variables are related to microscopic variables. If
the
microscopic problem can
be
solved for
some special situation, the
solution
can then
be used to get
values of the
macroscopic variables.
This cannot replace the
need
to
postulate
constitutive equations, but
is
can
give
insights
into the types
of terms expected
to
be
important in
the
constitutive
relations.
The philosophy
taken here is that
whatever
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information
can
be gained from
averaging is
worthwhile,
and therefore we
present a generic
averaging
method,
and
its
results.
If the reader
fee-s
that
averaging is
unnecessary,
he
can
skip
this
section, and
assume
the equations
(40) -
(43) have
been
postulated,
along with
the
interpretations given
at the
end
of
this Section.
Averaging the
equations
of motion is
suggested
by the
averaging
approach
to turbulence
(see
Hinze 1959).
Time
and
space
averages
were
the
first
to appear (Frankl
1953, Teletov
1958).
These
averages
have
been refined
by weighting,
by multiple
application,
and
by
judicious
choice of
averaging
region.
The
highlights can
be found
in the
work
of
Anderson
and Jackson (1967),
Vernier
and Delhaye (1968),
Drew
(1971),
Whitaker (1973),
Ishii
(1975),
Nigmatulin (1979),
and
Gough (1980).
Statistical
averages
are most
convenient for
the rheology
work,
see
Batchelor
(1974) for
a
summary.
The application
of statistical averages
to the
equations of
motion is
straightforward;
the paper
of
Buyevich
and
Shchelchkova
(1978)
summarizes
the approach
nicely.
Let
( > denote
an averaging process
so
that if
f(z,t)
is
an
exact microscopic
field, then
(x,t)
is
the
corresponding
averaged
field.
We shall specify
shortly the
requirements
which
an average
should possess,
however,
for now
we merely suggest
that it
should be
smoothing
in
the sense that
no
details
appear
in the
averaged
variables. Some
examples
of
commonly
used
averages in multiphase
flow
are
the time
average:
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2(X,t
)
212
L 3+
21/
f(x',t)dxdx~dxl'
(6)
L
x
-
1
L x
2 -
-
L x-
L
1
2 2 2
3
2
where
L
is an
averaging
length
scale;
a weighted
space
average
3
(x,t) =
ff
g(x-x )f(x ,t)dx
(7)
3
R
where
ff g(s)ds
=
1,
and various
combinations of
averages
and/or
3
specific types
of
weightings.
Also
mentioned in the
literature
are
ensemble
averages:
N
(x,t)
=
f
(x,t)
(8)
n=1
n
or
5(x,t)
= f
f(x,t;w)dU(w)
(9)
where
fn (x,t)
or
f(x,t;w)
denote
a realization
of the
quantity f
over
a set of
possible
equivalent
realizations
9.
One
way
in
which
randomness
might
be
introduced in
a particular
flow
situation
is
by
allowing the
particles to
have
random
initial
positions.
The averaging
process is
assumed to
satisfy
=
+
(10)
=
11)
= c
af
< -
(12)
at,-
at
af a
=
(13)
ax.
x.
1
1
The first
three
of these
relations
are
called
Reynolds rules,
the
fourth is
called Liebnitz'
rule, and
the
fifth is
called Gauss'
rule.
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Some difficulty is
encountered when trying
to apply
the
average
to
the equations
of motion
for each phase.
In
order
to do
this, we
introduce
the
phase
function
Xk(x,t) which
is
defined
to
be
1 if x is in
phase
k at time t
Xk(x~t)
=
1-(14)
0
otherwise
.
We shall deal with Xk as a generalized function, in particular in
regard
to differentiating
it. Recall that a derivative of a generalized
function can
be
defined in terms of a set of test functions , which
axk
are "sufficiently smooth and
have
compact support. Then and
a
Xk
axk
are
defined
by
i
aXk
f
ax (x,t)O(x,t)dxdt =
- Xk(2,t) 21
(m,t)dzdt ,
(15)
RxR
a
R3XR
axk
f
xi
(x,t)*(x,t)dxdt = - x
X
(x,t)dzdt (16)
R3xR xi
R
3xR
k
ax.
(xtdt(6
It
can
be
shown
that
axk
a-+
viVX
0
(17)
a
i
k=
in the sense of generalized functions To see this, consider
ax
k
R + Vi'Vxk )Odt
R3XR
f X
xk Lt
v.vdxdt
R
3xR
ka
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a+
Ov)dx)dt
f
(dJ
- .v
-
dt
R Wt)
dx)dt
dt
k
-
x 0_---o8)
since
has compact support
in t. Here
Rk(t) is the
region
occupied
by phase k at
time t, and
we
assume
that V
i
is
extended
smoothly through phase k
(in
order
that the
second
line makes sense).
If f
is
smooth except at
S,
then
fVXk
is
defined
via
f R3RfVXkdxdt
=
- fR
xkV(fo)dzdt
= f c fR
k
W V(fo)dxdt
= f
fs
nk
f
dSdt
, (19)
where n
k
is the
unit normal
exterior to phase k,
and fk
denotes
the
limiting value of f
on the
phase-k
side
of S.
It
is also clear
that Vk is zero, except at the
interface.
Equation
(19)
describes
the
behavior
of VX
k
at the
interface.
Note
that
it
behaves
as a "delta-function", picking
out the
interface
S,
and has the
direction
of the
normal interior
to phase
k.
This
motivates
writing
ax
VX
k
(20)
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where aX/an
is
a
scalar
,alued
generalized function
which
has
the
property that
fR3x
R
3n
x,t)] x,t)dxdt
=
0_
f
(x,t)dSdt 21)
The quantity
ax/an then picks
out
the
interface
S.
We
write
ax =
s
(22)
where
s
is
the
average interfacial
area
per unit volume.
Averaged
Equations
In order
to
derive
averaged
equations for the
motion
of each
phase,
we multiply the
equation
of conservation of
mass valid in
phase
k (1)
by
Xk
and average. Noting
that
aP
a axk
a
=
-
+
*
iV.5xk
(23)
and
X
k
V.pv =
V.xkpv - PvVxk
,
(24)
we have
t
+
V'
=
25)
Similar considerations
for
the
momentum equations
yield
a
W-
+
. = V.
+
(26)
k
+
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are
the interfacial source terms.
As
noted,
VXk
picks out
the
interface, and
causes
discontinuous quantities
multiplying
it to
be
evalued
on the phase-k
side of
the
interface.
The
jump conditions are derived
by
multiplying equations
(2.3)
and
(2.4)
by ax/an,
and recognizing
that
n = -2
.
We obtain
2 2
I
c[p(vv.Hkvxk>
=
I
rk =
0
(29)
k=1 k=1
2
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There
are two types
of
averaged
variables which
are
useful
in two-
phase mechanics, namely the
phasic, or Xk-weighted
average, and the
mass-weighted
average.
Which is
appropriate is suggested by
the
appearance of
the
quantity
in
the
equation of motion.
The phasic
average of the
variable
0
is defined
by
k
=
/ak (34)
and
the mass weighted
average of the
variable P
is
defined
by
A
4k =
/akpk
(35)
It
is convenient
to write
the stresses T
k
in
terms
of
pressures
plus extra
stresses. Thus,
T
k
=
-pkI tk
(36)
It is
expected that
readers familiar
with
fluid dynamical concepts
are
familiar with the
concept of
pressure in fluids; in
this
case, Pk
can
be
thought
of
as the
average
of
the
microscopic pressure.
If one
of
the
phases
consists of solid
particles, the
concept
is
less
familiar. In
this
case, the
microscopic stress (involving
small
elastic deformations,
for
example)
is
thought of
being made
up of a
spherical part (acting
-qually in all
directions)
plus
an
extra stress.
The
spherical
part,
when
averaged,
yields
the
pressure Pk
in equation (36).
It
has
further become
customary to
separate various parts of the
interfacial momentum
transfer
term.
This
is
done by defining the
interfacial velocity of
the
k
th
phase by
k.
r
kk,= ,
(37)
and the interfacial
pressure on
the
kth phase
by
Pk,iIVak
12
, 7
k
(38)
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Equation
(38) is the
dot
product
of
Va
k
of the standard
definition
(Ishii
1975)
of the
interfacial
pressure. The standard definition
uses
three equations
to
define one scalar quantity,
and cannot
be a
generally
valid
definition. Here the remaining
part
of
the contribution of
the
pressure
at the interface is
lumped
with
the viscous
stress
contribution
at the interface, and is treated through
the
use
of
a
constitutive
equation.
Thus, we write
Kk =
i
- Pk,i k k
4
d (39
d
k k
VXk
-
tVXk>
is
referred
to
as
the
interfacial
force
density,
although
it does not
contain
the effect of the
average
force on the interface due to the
average interfacial
pressure. The
term
-pk,i
Vk,
which does contain
the
force
due to the
average
interfacial
pressure,
is sometimes referred to as the buoyant force.
The
reason for this terminology
is, of
course, that the buoyant force on
an
object
is due to the distribution
of
the pressure
of
the surrounding
fluid
on
its
boundary.
With
equations (31) and (34) - (39), the
equations
of motion
(25)
and (26) become
a
+
Vkkk
k
(40)
aap
-
v
k-
-a
p
+*
(= +a
at k
kvvk
k kVPk k (t
+ k
+
kk,i +
(pk,i - k)V
k
d
(41)
+ q
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The jump conditions
(29) and (30) are
2
Z
r
k
=
o
42)
k=1
2
2 [
r ~~ +p
1
d
=.
43)
k1 k k,i +pk,i
k
+
PO Mm(3
=
1
Pkikm
Adequate models for
compressibility and phase
change require
consideration
of
thermodynamic processes.
These are beyond the scope
of
this paper;
therefore we shall
restrict
our
attention
to
incompressible
materials
where
no phase
change occurs.
Thus we assume that
k
=
constant
(44)
and
r = 0
.
(45)
k
In
order to
simplify
the
notation, we shall
drop all
symbols
denoting averaging.
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CONSTITUTIVE
EQUATIONS
In
order to
have a useable
model
for
an
ordinary
single-phase
material, relations
must
be
given
which
specify
how the
material
interacts
with itself. These
relations
specify how the
material
transmits forces from
one part
of the body
to
another. For single-phase
materials
a relation between
the
stress and
other field
variables is
usually required.
For
two-phase
materials, where we
desire to
track
both phases,
we must
specify
the stresses for both
phases and
the way in
which the
materials
interact across the
interface. Thus, we require
constitutive
equations
for the
stresses (T
+
0
k), the
interfacial
force
density
d, and
the pressure
differences
pk
- Pk,i'
consistent
with
the equations
of
motion and the
jump conditions.
The
fundamental process
consists
of proposing
forms for the
necessary
terms
within the
framework of the
principles
of constitutive
equations, finding
solutions of
the
resulting
equations,
and
verifying
against
experiments. The
process can
be
iterative,
with
some
experiment
indicating that
a
more involved
form is needed
for
some
constituted
variable. The
ideal end result
of
the process is
a set
of equations
which
could be used to
predict the
behavior of the
two-phase
flow, for
example with
a computer code.
With
the equations
should come
a set of
conditions
for the
validity of
the values of
the constants and
other
functions
used
in the
constitutive
equations.
The stresses
Tk +
ak,
the
interfacial
force density
d and the
pressure differences
p- pki
are assumed to
be
functions of a
a
k/at, Vak,
Vk,
VVk,
3Vk/at
...
where
...
represents
the
material
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properties, such
as
the
viscosities
and
densities
of the
two materials,
and
other
geometric
parameters such
as
the
average particle
size, or the
interfacial
area density.
For
concreteness, we shall
refer
to
phase one as the
particulate,
or
dispersed
phase and include in that
description solid
particles,
droplets,
or
bubbles.
Phase two is then the
continuous,
or carrier
phase,
and can be
liquid
or
gas. We
shall denote
=
a
(46)
so that
1-a a
(47)
2
It
is evident
that both
a and 1-a
need not
be
included
as
independent
variables
in
forming
constitutive
equations.
Drew and
Lahey
(1979)
consider the general
process of constructing
constitutive
equations. The
principle of consequence
is
that of
material
frame indifference,
or objectivity. This
principle requires
that
constitutive
equations be appropriately
invariant
under a change of
reference
frame. The motivation
behind this
assumption is
that we
expect the
way in which
a
two-phase
material distributes
forces to be
independent of the
coordinate system
used
to
express
it.
Those who
doubt this
principle
often argue that the
momentum equation is
not frame
indifferent,
needing
Coriolis forces
and
centrifugal forces to
correct
it
in
non-inertial frames.
This argument is
incorrect, since the
appropriate
formulation of the
momentum
equation deals not
with
"acceleration"
but
with
"acceleration
relative
to
an
inertial
frame .
When formulated
with the
"acceleration
relative to an
inertial
frame ,
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the
momentum
equations
are
frame
indifferent
(Truesdell
1977).
This
introduces
an
interesting
question:
Can
the
mixture
know
whether
or
not
it is being
referred
to
a inertial
frame?
That
is, can
the
interfacial
force
and
the
Reynolds
stresses
depend
on the
reference
frame?
The
principle
of
material
frame
indifference,
as used
here,
claims
that
they
cannot.
Others
claim
that
they
must (Ryskin
and Rallison
1981).
The
principle
of
objectivity
is
now
examined
(Drew
&
Lahey
1979).
The
approach
we take
deals
with
coordinate
changes,
but is equivalent
to
more abstract
approaches
(Truesdell
1977).
Consider
a
coordinate
change
from
system
z
to
system z
,
given
by
=
Q(t).z
+
b(t) ,
(48)
where
Q(t) is
an
orthonormal
tensor
and
b is
a
vector.
A scalar
is
objective
if its
value is the
same
in
both the
starred
and
unstarred
systems,
that
is,
if
0
(X
,t) =
xt) .
(49)
A
vector
is
objective
if it
transforms
coordinates
correctly,
that
is,
if
v
= Qv
.
(50)
A tensor
is
objective
if
*
t
T
= Q-TQ
51)
The
scalar
a
is objective;
however
Da/3t
is
not.
It
is
straightforward
to
show
that
D /Dt
=
a/3t +
Vk
-Va
is
objective.
Let
us next
consider
objective
vectors.
The
volume
fraction
gradient
Va is
objective.
If
we
differentiate
(48)
with respect to
t
following
a
materitl
particle
of
phase
k,
we see
that
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=
Q(t)k
+
)
(52)
Hence
velocities
are
not objective. This
is obvious
physically, since
the
velocity
of
a point
depends
on
the
coordinate system.
If
we
take
equation (2.53),
for k 1
and
2, and subtract, we
have
V
1
-
2
=
Q (vl
- 7
2
)
(53)
Thus, the relative
velocity between the phases is objective.
For acclerations,
differentiating
(52) following
a material
particle
of
phase
j yields
Dv
.v
QO
+ Qx
+ Qv.
+ Qv
+
b .
(54)
Dt Dt
k
Thus, subtracting
the
value
for j =
1, k = 1 from
the value for
j = 2, k =
I yields the
result that
DI
2
Dv2
12
t
Dt
is objective.
The
list
of tensors which
we consider
is motivated
by the
desire to
model two-phase
flows. Thus, velocity
gradients
(expressing rate
of
deformation)
are included,
but displacement gradients
(expressing
deformation)
are not.
It is well known
(Truesdell and Toupin
1960)
that
=
-
(V +
(VVk) )
(56)
Dk,b
2 k
+
are objective.
We
further note that
V(,
I
-
v
is
an
objective
tensor.
Thus,
the
problem
of
forming
constitutive equations
reduces
to
expressing
Tk +
0
k' k - Pk,i'
(57)
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in
terms
of
a,
Dka/Dt,
Va,
v - v
2
, D
2
vI/Dt
-
D
1
v
2
/Dt,
Dib, D
2
,bV(VI - v
2
)
. (58)
We
shall assume that
the materials are
isotropic.
This
means
that
no
direction is preferred
in either
material or in the
way they
interact. Note that
a
particular flow may occur
in such a
way that
a
preferred
direction (down,
for example)
might
emerge.
In
that case, the
flow is
anisotropic, but the materials
and
the mixture
are isotropic.
If
f
is
a
scalar,
then
f
can depend
on the
scalars and
the
scalar
invariants
formed from the vectors and
tensors in
the list
(58). The length of
a vector is
a
scalar
invariant,
as
are
the
trace,
determinant
and Euclidean
norm of
a tensor.
If
F is a
vector, then
F
must be
linear in all
the
vector
in
the
list
(58),
plus any
vectors
which can be
formed
in
an invariant way
from the
vectors and
tensors in that list.
Thus
F
=
a. V.
(59)
1
1
where
V
i
are the
objective
vectors
so
formed.
Each scalar
coefficient
a
i
can be
a function of all
the scalars
and scalar invariants.
If F
is a second
order tensor, then
F = E B .
(60)
j J
where
Y. are the
objective tensors which can be
formed
from the
list
I
(58).
The
scalars B. can be
functions of all the
scalars. If F
is
symmetric,
we
need only
consider
symmetric
Y..
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The general
approach
(Drew and Lahey 1979)
gives a problem
too big
to
be
practical
or
meaningful.
Thus,
we shall
consider
here
forms
for
the
stresses,
etc.,
which
we expect
to
be
important in two-phase
flows.
Obviously,
we shall
have
the constant
worry that something
has
been forgotten; however,
the alternative
of
stopping at
this point,
awaiting evaluation of
over two-thousand scalar
coefficients
seems
less
attractive.
Stresses
Let
us now
discuss
specific
considerations for the specific
constitutive equations needed.
Let
us
begin with the
stresses.
The
extra
stress
(viscous
or
elastic) and turbulent stresses are
combined as
T
k
+ a
k .
Macroscopically,
at least in
the context of this model, one
sees no
way
to
separate them. Microscopically,
one
is
due
to
the actual
stresses, and the
other
is
due
to
velocity fluctuations from the mean.
In many
cases
of
interest
the extra
stress
is
smaller than
the
turbulent
stress,
one
reason
being
that
any
appreciable slip between the phases
generates
velocity fluctuations,
which appear in this
model
in
the
turbulent stresses.
These fluctuations can
transport
momentum.
The
conceptual situation is analogous
to
that for
single-phase turbulent
momentum transport, with
eddies carrying momentum
across
planes
parallel
to
the
shearing direction
and
losing it into
the
mean
flow.
The
fluctuations in single-phase
turbulent flow are
there because of
instabilities in the
laminar shear
flow;
in two-phaqe flow
they also
come from fluctuations
generated
due to
the flow
around individual
particles.
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Drew and Lahey
(1981)
have generated
a model for
the
turbulent
stresses
in
bubbly
air-water
flow. The model
they deduce in
this
case
has the form
0 =
2u2 D
+ a I +
b2(
I
-
V
)(V -
v
2
)
61)
2 2 2,b 2 2 1 -2
1 2)
T
for the
liquid
phases. Here v2 is
the
eddy viscosity, a
2
is an
22
induced eddy pressure ,
and b
2
is
associated
with
bubble passages.
T
The
parameters p
2
, a2 and b
2
depend on a,
the
bubble radius
r,
the
relative velocity
IVl -
w
2
I
and the
Euclidean
norm
of
D
2
,b.
Nigmatulin (1979) uses
a
cell
model, assuming inviscid
flow around a
spherical
particle in
each
cell,
to calculate
values
of a
2
and b
2
.
He
obtains
b =-
(62)
2
2 2
a2 IV
- (2
(63)
a
2
-
6
2 1
2
He does
not obtain a term
involving
the
eddy viscosity
because
the
cell
model
is unsuited to obtaining
shear
flow
results.
His results
are
valid
for
dilute
flows.
Other
effects
are
present
in
most turbulent
flows.
Drew
and Lahey (1981) used
mixing lengths to
evaluate the
coefficients in
eq.
(61).
The data
available
thus far
(Serizawa 1975)
allows
only
evaluation
of the ratios of
various Karman
constants,
nonetheless, the
importance of
the
problem
in
nuclear
reactor
technology
has spurred the
acquisition of more
direct
data
in
special cases
(Lance
1981).
This
data should
be
available shortly, and should allow
the
evaluation
of
a
2
and
b
2
for
bubbly flows.
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Evidence (Serizawa 1975) also
indicates
that
a
=
O(
2
2
(64)
and is negligible in dispersed bubbly flows. A model for 0 analogous
to
eq.
(61) is
a=
21TDI
aI+ b v
- v2)(v1 - v)
(65)
1 1,b
1 1 1
If the particulate phase follows the fluid phase closely, the quantities
in
equations (61) and (65) representing both velocity scales and mixing
lengths of large eddy processes should be approximately equal. If this
is
the
case,
then
T. P1
T
P1
p
2
2(66)
The remaining
terms in eq.
(65) arise
at least in
part
due
to
the
motion of the particles through the
fluid.
No constraints are placed on
these coefficients.
Ishii (1975) argues that for the
viscous
fluid phase,
the
average
of the microscopic viscous stress lead to
T
=
2j2
D
+ P
b(1-a) EVQ(V - V2)
+ (V
- V
2
)V
u
] (67)
2 2 2,b
2
1-a
1 72 1 - 2)a
where P
2
is
the exact
viscosity
of the
fluid, and
b is called the
mobility
of
phase
two.
He argues that
b(1-a)
1
for a near
zero.
No experimental evidence
has
been offered
to
verify
(67).
In the case when the
particulate phase consists
of
solid
particles,
it is
usually
assumed that
T
= 0.
If the particulate phase is
also
a
viscous fluid, then it is sometimes assumed that
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T
I
=
2P1D1,b
(68)
This corresponds
to
Ishii's result (see
eq.
67))
with
b = 0.
Murray
(1965a)
offers
a
model
for a particle-fluid
flow in which he
assumes
that
the
particles
experience a viscous stress with a viscosity
coefficient
proportional to the fluid viscosity.
His
argument indicates
that
the force he is calculating is not a
stress, but an
interfacial
force.
Pressure Relations
The pressure differences
pk
-
Pk,i
are
discussed
next.
First, we
note that with
the
assumption of
local
incompressibility of each
phase,
we
must
constitute
all but one of P
1
1 P
2
1 Pl,i
and P2,i. The
other
will then be a
solution
of the
equations
of
motion.
The simplest assumption
for
pressure differences,
is to assume
none. That is, assume Pk =
Pk,i
for
k = 1,2. This assumes that
there is instantaneous
microscopic pressure equilibration which
will
be the case if the
speed of sound
in
each phase
is
large compared with
velocities of
interest.
In applications
which do not deal with acoustic
effects
or
bubble expansion/contraction, this assumption is adequate.
For a
discussion
of
bubble oscillation effects, see
the
article
by van
Wijngaarden (1972). Cheng
(1982)
studies the consequences of several
modeling assumptions on wave propagation in a
bubble
mixture.
In
situations where
surface tension is
important and no contacts
occur between
the
particles,
we assume that the
jump condition
(4)
becomes
1 2
p -p
-O K ,
(69)
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where K'
is
the
exact curvature
of the interface.
Multiplying
by
VX1
and
averaging
gives
(P1,i -
P2,i
)V
a = aK V
(70)
where
K is
the
average mean curvature. Since this equation holds
for
all
Va,
we
have
Pl,1
- P 2,i
= OK
.(71)
If
the
surface
tension
is
negligible,
equation (2.72)
gives pl,i
=
P2,i
When
contacts occur between
particles, these
considerations
are not
always simple. The
contacts provide mechanisms for causing the average
pressure at
the
interface in one
phase
to be
higher
than
the other.
Consider solid particles
submerged
in fluid, with no
motion, and with
the
particles
under
a compressive force (supporting
their own weight, or
the weight
of
an
object, for example). The resulting
stress will be
transmitted from particle to
particle
through
the areas of
particle-
particle
contact.
Just on
the
particle
side
of
the
interface where
the
contact
occurs,
the stress
(pressure) level will
be
large. At places
where
no
contact occurs,
the stress
will be equal to the pressure in the
surrounding fluid.
Under
normal circumstances,
the contact areas will
be
a
small fraction of
the total
interfacial
area.
Thus, the
approximation pli p
2
,
p.
is
usually valid. In the
more
general
situation, these contacts may be
intermittent, when the
particles
are
bumping
together,
or
may
be
constant,
as
when
the
particles
are
at
rest. In
this case, also,
we
expect P1,i
=
P2,i Pi
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stuhmiller
(1977)
argues
that
p
2
- P2,i
=
P
2
IV
2
-
V
12
(72)
where F
is taken
to be
a
constant. The
argument
leading
to
(72)
is
exactly the
calculation
of
form drag,
which is
usually
part of
the
interfacial
drag.
It
is not
clear
whether
using (72)
and
a
drag law
(79)
includes
the
form
drag
consistently.
We shall
assume that
for
the
fluid
phase,
P
2
= pi. To
allow for
the
possibility
of
the extra
stress due to
contacts,
in the
particulate
phase
when it
consists
of
solid
particles,
we
shall write
PI = Pi,i
+ Pc = P2
+ Pc
(73)
where pc
is the
pressure in
the
particles due
to
contacts.
A model
is
needed
for
PC.
Several modelers
(Gough
1980,
Kuo
et
al.
1976)
have
used
a model
with Pc
= pc(a),
with Pc
(a) =
0
for
a
< ac, where
ac is
the
packing
concentration
of
the
particles,
and
pc (a) rapidly
increasing
for
a
> a
.
This
further
suggests
an "incompressible"
c
model,
with
Pc
= 0
for
a
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the
momentum
jump
condition becomes
Kd
+ V +
d- p V
= oK
Va
* (75)
+
Pli
M2 - 2,i
Using
condition
(71),
we
have
+d =
0
(76)
1
2
Therefore, we need only provide a model for d. K then is determined
1
2
by the jump condition (76).
The
interfacial momentum
transfer
1
contains the forces
on
the
particulate
phase
due
to
viscous
drag,
wake and boundary layer
formation,
and
unbalanced
pressure
distributions
leading to
lift or
virtual
mass
effects, except
for the
mean interfacial pressure.
The
model which
we
discuss here
should
contain,
as much
as
possible,
the
above forces.
Indeed, this
motivates
the inclusion
of
certain
quantities
in
the list
of
variables
(58).
We postulate
da
1
2
v
Kd
=
A v
-
v)
+
A
2
3-
1 2 1
2
(T --
+
v
1
VV
2
) (---
+
v
2
Vv
1
+ A3(v
2
- v1)D + A(v -
v)'D
+ A(v -v)V(v
2
- )
32 1 1b
4(2
1
2,b 5(2
1 2
+ A6
V - v2).[V(vl - v2)]t
(77)
where
A, - A
6
are scalar functions of the invariants.
The first
term
represents the classical drag forces.
To conform
with customary useage,
we
write
C
A 1
P2
1
v1
- v21
(78)
where r
is
the
effective
particle
radius,
and
CD
is
the drag
coefficient.
It is
usually assumed
that CD
=
CD (a,Re), where
0
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Re
= 2P
2
IV
- v
2
r/p
2
is
the
particle Reynolds number.
A
careful
study, including extensive comparisons with data,
is
given
by Ishii and
Zuber (1979).
Their
conclusions
for
the drag
coefficient are
summarized
in
Table 1, which
is adapted from
their paper.
The
combination
av
2
av
1
A2[(-
+ V.VV2) - (---l
+
V2 VVl)] +
A
(V
- V) -
v()
(79)
2 at
1
2
at
2 1
5 2
1
*~2
1
is
an
acceleration
term.
Drew,
Cheng
and
Lahey (1979)
write
A =a PC
M(a)
,
(80)
A
= A
2(1-)(a))
,
(81)
where
C is
referred to
as the
virtual
volume. If
CVM =
and no
spatial
velocity
gradients are
present, then (79)
reduces to the
classically
accepted virtual
mass force.
It
is
less
easy to
obtain
correlations
for CVM than for
C
D
, since the ratio
of virtual
mass
forces
to drag
forces
is
of
order
V t
/r,
where
V is
a velocity
scale and
t
is
a
time scale.
In
order for
the virtual
mass force to
be
appreciable,
the
time scale must
be of
the order r/V. For
most
multiphase
flow
applications,
r
is small,
and
V is not.
Thus,
we
see virtual
mass
effects
only
at
relatively
high frequencies.
Zuber (1964)
considered
finite concentration
effects
on virtual
mass by
considering
a
sphere
moving
inside
a spherical
cell . He
obtained
C
(a)
- 1 1+2a
1 +
3
(82)
VM = 2 --
=2
for
small
a.
Nigmatulin (1976)
also calculates
the virtual
mass
coefficient from
a
cell
model. He
obtained
C
(a) -
1-
to 0 0).
The
VM = 2
form
of
the
acceleration
which he derives
is not
objective.
It
is
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possible that
this
is due
to
the inability
of
cell models
to
adequately
deal with
velocity
gradient
terms.
The
value of the
virtual
mass
coetficient
derived by van
Wijngaarden
(1976)
is
C (a)
=
0.5
+ 1.39a .
(83)
VM
Moheyev (1977)
obtained
an empirical
correlation
C M(a)
=
0.5
+
2.1a
(84)
by using an
electrodynamic
analog. Thus,
there seems
to
be a
concensus
that
C + I
as
a +
0.
There
is
less
agreement
on the O(a)
VM 2
correction.
In
addition, there
are
several models
in
the
literature
(Wallis 1969,
Hinze
1972)
which use
non-objective
forms of the
virtual
mass.
There is
no
compelling
evidence
at this
point as
to
whether the
objective
or
non-objective
forms
conform more
closely to
reality.
Cheng (1982)
has
done
extensive
predictions
using
virtual mass
models for
accelerating
air-water
bubbly
flows. For low
frequency
small
disturbances in
a
one-dimensional
flow, he
found
that the
value
of
CVM
had a
significant
effect on the
propagation
speed,
but
that the
data
were
scattered so
that
a particular
value could
not be
estimated.
For higher
frequency
waves (sound
waves)
and for
critical
flow,
compressibility
effects were
important. For
nozzle
flow, the
virtual
mass effects
were
insignificant.
None
of
the calculations
showed
any
effect of
different
values of
X.
The
term
A4 (2 -
v1)-D2,b
(85)
contains the
effect of the
lift.
We
write A
= a L. On
the other
hand,
the forces
represented
by A3
(V
2
-
vI) D and
32aw-1
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t
A (V
-
V2 ).[V(V
-
v
2)
have no
analogs
in single
particle
calculations, and, no
observations
confirm
their presence.
Thus
we
assume A
3
= A
6
= 0. Very little
experimental
evidence for A
4
exists.
Interfacial
Geometry Models
It is
evident from eqs. 71) and (78) that
a relation
is
needed for
expressing the
geometry of the
interface. Several such relations
have
been used;
however, none is so
compelling as to be
called general. The
one
which comes closest
is
due to
Ishii (1975)
(derived for
the
time
average; the
generalization
is
obvious). We shall
discuss this later.
First, we
present the
simplest
models.
If
the
particles are solid
spheres each of
the same
radius
r, we have
4 3
a
=
-y
r
n
(86)
where n
is the
number density.
Other quantities
of interest are
immediately
obtainable
from
a
and
r, for example,
the average
3a
interfacial
area per unit
volume is
---. Note that n is
conserved
r
since a is;
specifically, we
have
an
+ V-n
v
1
=
0
. (87)
If the
particles are not
monodisperse, it may be
necessary to derive
a
model where
each
size
of
particle is treated
separately.
Some recent
efforts along these
lines
show
great
promise
(Greenspan 1982).
Particle
breaking,
agglomeration, or
accretion/erosion
also
necessitates
a more
complicated model.
An
effort (without macroscopic
mechanical
considerations)
is
given
by Seinfeld
(1980).
Both
approaches
are
essential
geared to an
equation analogous
to
(87)
for a number
density.
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A more
general approach
is suggested
by Ishii
(1975) (for time
averaged equations), where he writes
as
O
at
+ vs =
i
(88)
where
s
is the
interfacial area per unit
volume, which
in
the
present
notation
is , v.
is
the
averaged interfacial velocity, and
*i
is a source
of
interfacial area, due to
bulging of
the
interface.
This
approach is probably
best suited
to
flows
where the
particulate
phase
(bubbles or drops) can
coalesce or break up. It has not
been
investigated to any
degree.
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MODEL CONSIDERATIONS
The
main aspect of
two-phase
flows which
makes
them
fundamentally
different from
a single-phase flow
is
the
interfacial
interaction terms,
and,
thus, in the absence of
interfacial mass transfer
the
difference is
d
contained
in
the
interfacial
force Nk. The
modeling of this term
must
proceed cirefully,
with attention paid to
any
possible
inconsistencies
or
untoward
predictions.
A canonical
pioblem involving simplified two-
phase
flow models has
surfaced in the
literature;
namely, that the
one-
dimensional,
incompressible,
inviscid
flow equations
without virtual
mass effects are
ill-posed as an
initial
value problem (Ramshaw
& Trapp
1978).
To see this, consider
the equations
with
CVM = 0 and L
= 0.
(This
represents
the
model
used in most
practical
calculations.)
aa
3a
+ ---
= 0
(89)
-
+
2
= 0
(90)
t , ax
ap j v 3-Vl1)
a
p+A v
91
p1 (Y-
+ =w - - + A
1 2
- v
1 )
(1
av
av
a2
2 ap
(1-a)p
2 --
+ v
2
=
-a-) ;
+
A (V
1
- v
2
) (92)
If
the
pressure is
eliminated
from
system (89)
-
(92)
and the
resulting equations
are
expressed
in
the
form
A-2H
+ B
2H
=
C,
where
t
u = [,v
1
,v
2
] , A and
B are
3 x 3
matrices
and
C is a 3-vector,
dz.
then the
characteristics
of the
system are given
by
,--Vi,
i
= 1,2,3,
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where V. are the three
characteristic
values of
det[AV-B] = 0.
I
Note that since
C
contains
no
derivative
terms,
it is
not
involved in
determining
the
characteristics.
The
characteristic
values V. for the
system
as
given
are a
pair
of complex conjugate
roots
V1
= a iO
(with
0
yk ), and V
3
= 0
(resulting
from the assumption of
incompressibility).
This
implies (Garabedian
1964)
that
the
initial
value problem is
ill-posed.
One
implication
of the
ill-posed
nature of the
system
can be seen
as follows. Consider the
linear stability
of
a constant solution.
The
solution
of the linearized equations
has the form u(x,t)
=
Vt+ikx
u
0
+ u
1
e
, where k
is
real. Substitution into
the
linearized
system results in the
eigenvalue problem
[A(u0 )v + B(u
0)ik
-
C(u
0)u
= 0 (93)
for V. The
eigenvalue equation
is
C(u
0
)
det[A(u - -
B(u
----
1 = 0
o
(94)
0 k
0
k
For
k large, two roots of (94) are
vi
i
(95)
Thus,
V
k(8 -
ai)
*
(96)
1,2
As long as 0
, 0,
one of the
v's will have positive real part,
indicating linear instability. Note, too, that as k increases, the
growth
rate lOkI increases.
This
shows
that finer disturbances
grow
faster. This implies
that at any
t
>
0, maxlule
t+ikx
I
can
be made as
large as
desired by taking k sufficiently small.
Contrast
this with
the
behavior of
an
instability of a well-posed system, where
the
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vt~ikx
solution u e
can be
made
as small as
desired
at
a
finite
t by
making
u
1
small,
for
any k. The
instability in
the
well
posed
system
has
some
realistic
physical
meaning, while the
instability
always
present in the
ill-posed
system suggests
that
the model
is not
treating
small scale
phenomena correctly.
The
extension of
the
argument
to
three space
dimensions
is
straightforward, and
shows that the
system is ill-posed in
three
dimensions also.
The
ill-posed nature of
the model
makes it
desirable to
seek
physical
modifications of that
simple
model to
find a well
posed
model.
An effort has
been made
to
determine
whether
and/or to
what
extent
virtual
mass makes the
inviscid model
well
posed.
Lahey,
et al.
(1980)
have
shown that
the
model becomes
well-posed for
certain
values
of
CVM. The value
of
CVM which makes
the
system well-posed
is
usually large
unless
a
is extremely
small, or
other
somewhat non-
physical assumptions
are
made.
Thus,
even
though
it is
a
possible
fix
to the question
of
the
well vs.
ill-posed
nature
of the
model,
virtual
mass
does not seem
to be the
total
answer.
We note here
that
if the
viscous and
Reynolds stresses
are
included
in the
equation, and they
are
assumed
to
depend on
Vvk,
the
equations
become
parabolic,
and hence
are
well-posed
if
enough
boundary
conditions
are used.
Also
Prosperetti
and
van Wijngaarden
(1976)
have
shown that
using
certain compressibility
assumptions in
gas-liquid
systems
gives
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real characteristics.
Stuhmiller
(1977)
shows that
his model
including
the
form drag (see
eq.
72)
gives real
characteristics
for
sufficiently
large
F.
The
question of the
role of the
inviscid
model
remains. In
single-
phase
fluid
mechanics,
the inviscid
model
is
extremely
important,
and
governs
the
flow
outside of
boundary
layers,
shear layers
and
oth